Jump to Any Term

A sequence can hide a shortcut. You could hop along twenty-five boxes, one by one... or uncover a rule that lands on the answer in seconds. Watch the numbers. Their pattern is about to turn into a machine!

A linear sequence changes by the SAME amount each time. Here are five, eight, eleven, fourteen. Each arrow adds three. That equal jump is the clue. The term number tells you the position: first, second, third, fourth. The term value is the number sitting there. Now connect position to value. Multiply the term number by three, then add two. For the fourth term, three times four is twelve. Add two, and fourteen appears. The rule, three times the term number plus two, works for every position. Order matters. If a rule says two plus three times the term number, do the multiplication first. Do NOT work automatically from left to right. Keep the term number in the input slot, finish the multiplication, then add two.

Let's work through an example. A sequence begins seven, eleven, fifteen, nineteen, and continues with the same rule. What is the twenty-fifth term? First, inspect the gaps. Seven to eleven is plus four. Eleven to fifteen is plus four. The constant jump is four, so the position is multiplied by four. But four times the first position gives four, not seven. We need three more. So the rule is: multiply the term number by four, then add three. Check it with the fourth term. Four times four is sixteen. Add three... nineteen. The rule fits. Now place twenty-five into the rule. Four times twenty-five is one hundred. Only after that multiplication do we add three. One hundred plus three is one hundred and three. So the twenty-fifth term is one hundred and three.

Remember: find the equal jump, build a rule connecting position to value, then substitute the named term number. Follow the operations in their stated order. You are not trudging through a sequence now - you are jumping straight to any term!